Exponentials and Logarithms

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A bacterial culture starts with 1000 bacteria and doubles every 3 hours. Simultaneously, a disinfectant reduces the population by 20% every hour (i.e., the population is multiplied by 0.8 each hour). After how many hours does the net population reach 5000? Provide the exact answer in terms of natural logarithms.

t = \frac{3\ln 5}{7\ln 2 - 3\ln 5}

hours

t = \frac{3\ln 5}{7\ln 2 + 3\ln 5}

t = \frac{3\ln 5}{7\ln 2 - \ln 5}

t = \frac{\ln 5}{\ln 1.6}

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The concentration of a certain drug in a patient's bloodstream (in mg/L) after $t$ hours is modeled by $C(t) = e^{2t} - 5e^{t} + 6$ . At what times $t$ is the concentration zero? Solve for $t$ .

t = \ln 2

and

t = \ln 3

t = 0

and

t = 1

t = 2

and

t = 3

t = \ln 5

and

t = \ln 6

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Solve for $x$ : $3^{2x+1} = 5 \cdot 2^{3x-2}$

x = \frac{\ln(12)}{\ln(9/8)}

x = \frac{\ln(5) - 2\ln(2)}{\ln(3) - 3\ln(2)}

x = \frac{\ln\left(\frac{5}{12}\right)}{\ln\left(\frac{9}{8}\right)}

x = \frac{\ln(15/4)}{\ln(9/8)}

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Exponentials and Logarithms

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